Real Analysis with Economic Applications - Content Preface pps

. REAL ANALYSIS with ECONOMIC APPLICAT IONS EFE A. OK New York Universit y December, 2005 . mat he matics is very much like poetry what m akes a good poem - a great poem - i s that there is a large amount of thought expressed in very few word s . I n t his sense formulas like e πi +1=0 or ] ∞ −∞ e −x 2 dx = √ π are poems. Lipman Bers ii Con ten ts Preface Cha p ter A Prelim inaries o f Real Analys is A.1 El emen ts of Set Theory 1 Sets 2 Relations 3 Equivalence Relations 4 Order Relations 5 Functions 6 Sequences, Vectors and M atrices 7 ∗ A Glimpse of Advanced Set Theory: The Axiom of Choice A.2 Real Numbers 1 Ordered Fields 2 Natural Numbers, Integers and Rationals 3 Real Numbers 4 Intervals and R A.3 Real Sequences 1 Convergent Sequences 2 Monotonic Sequences 3 Subsequential Limits 4 Infinite Series 5 Rearrangements of Infinite Series 6 Infinite Products A.4 Real Functions 1 Basic Definitions 2 Limits,ContinuityandDifferentiation 3 Riemann Integration 4 Exponential, Logarithmic and Trigonometric Functions 5 Concave and Convex Functions 6 Quasiconcave and Quasiconvex Functions Chapter B Coun tability B.1 Countable and Uncoun table Sets B.2 Los ets and Q B.3 Some More Advanced Theory 1 The Cardinality Ordering 2 ∗ The Well Ordering Principle iii B.4 Application: O r d in al U tility T he ory 1 Preference Relations 2 Utility Representation of Complete Preference Relations 3 ∗ Utility Representation o f Incomplete Preference Relations Chapter C Metric Spaces C.1 Ba sic Notions 1 Metric Spaces: Definitions and Examples 2 Open and Closed Sets 3 Convergent Sequences 4 Sequential Characterization of Closed Sets 5 Equivalence of Metrics C.2 Connec ted n e ss and Separability 1 Connected Metric Spaces 2 Separable Metric Spaces 3 Applications to Utility Theory C.3 Co mpactness 1 Basic Definitions and the Heine-Borel Theorem 2 CompactnessasaFiniteStructure 3 Closed and Bounded Sets C.4 Sequen tial Compactness C.5 Co mpleteness 1 Cauchy Sequences 2 Complete Metric Spaces: Definition and Examples 3 Completeness vs. Closedness 4 Completeness vs. Compactness C.6 Fixe d Point Theory I 1 Contractions 2 The Banach Fix ed Point Theorem 3 ∗ Generalizations of the Banac h Fixed Point Theorem C.7 Applications to Function al Equations 1 Solutions of Fu nctional Equations 2 Picard’s Existence Theorems C.8 Products of Metric Spaces 1 Finite Products 2 Countably Infinite Products Chapter D Continuit y I D.1 Co ntinuity of Functions iv 1 Definitions and Examples 2 Uniform C ontin uity 3 Other Contin uity Concepts 4 ∗ Remarks on the Differentiability of Real Functions 5 A Fundamental Characterization of Con tinuity 6 Homeomorphisms D.2 Continuity and Connectedness D.3 Continuit y and Compactness 1 Continuous Image of a Compact Set 2 The Local-to-Global Method 3 Weierstrass’ Theorem D.4 Se micontinuity D.5 Applications 1 ∗ Caristi’s Fixed Poin t Theorem 2 Continuous Representation of a Preference Relation 3 ∗ Cauchy’s Functional Equations: Additivity on R n 4 ∗ Representation of Additiv e Preferences D.6 CB(T ) and Uniform Convergenc e 1 The Basic Metric Structure of CB(T ) 2 Uniform Convergence 3 ∗ The Stone-Weierstrass Theorem and Separability of C(T ) 4 ∗ The Arzelà-Ascoli Theorem D.7 ∗ Extension o f Continuous Functions D.8 Fixe d Point Theory II 1 The Fixed Point Property 2 Retracts 3 The Brou wer Fixed Point Theorem 4 Applications Chapter E Con tinuit y II E.1 Correspondences E.2 Continuity of Correspondences 1 Upper Hemicon tinuity 2 The Closed Graph Property 3 Lo wer Hemicontinuity 4 Continuous Correspondences 5 ∗ The Hausdorff Metric and Continuity E.3 The Maximum Theore m v E.4 Application: Stationary Dynamic Programming 1 The Standard Dynamic Programming Problem 2 The P rinciple of Optimality 3 Existence and Uniqueness o f an Optimal Solution 4 Economic Application: The Optimal Growth Model E.5 Fixed Point Theory III 1 Kakutani’s Fixed Point Theorem 2 ∗ Mic hael’s Selection Theorem 3 ∗ Proof of Kakutani’s Fixed Point Theorem 4 ∗ Contractive Correspondences E.6 App lication: The Nash Equilibriu m 1 Strategic Games 2 The Nash Equilibrium 3 ∗ Remarks on the Equilibria of Discontin uous Games Chapter F Linear Spaces F.1 Linear Spaces 1 Abelian Groups 2 Linear Spaces: Definition and Examples 3 Linear Subspaces, Affine Manifolds and Hyperplanes 4 Span and Affine Hull of a S et 5 Linear a nd Affine Independence 6 Bases and Dimension F.2 L inear Operators and Functionals 1 Definitions and Examples 2 Linear a nd Affine Functions 3 Linear Isomorphisms 4 Hyperplanes, Revisited F.3 A p p lication: Expected Ut ility Theory 1 The E xpected Utility Theorem 2 Utility Theory under Uncertainty F.4 ∗ Application: Capacities and the Sh apley Va lue 1 Capacities and Coalitional Games 2 The Linear Space of Capacities 3 The Shapley Value Chapter G Convexit y G.1 Co nvex Sets 1 Basic Definitions and Examples 2 Convex Cones vi 3 Ordered Linear Spaces 4 Algebraic and Relative In terior of a Set 5 Algebraic Closure of a Set 6 Finitely Generated Cones G.2 S epara tion an d E x tension in Linear Spa ces 1 Extension of Linear Functionals 2 Extension of Positive Linear Functionals 3 Separation of Convex Sets by Hyperplanes 4 The External Characterization of Algebraically Closed and Convex Sets 5 Supporting Hyperplanes 6 ∗ Superlinear Maps G.3 Reflection s on R n 1 Separation in R n 2 Support in R n 3 The C auchy-Schwarz Inequalit y 4 Best Ap proximation from a Convex set in R n 5 Orthogonal Projections 6 Extension of Positiv e Linear Functionals, Revisited Cha p te r H Econo m ic App licat ions H.1 Applica tion s to Expected Utility Th eory 1 The E xpected Multi-Utility Theorem 2 ∗ Knigh tian Uncertainty 3 ∗ The Gilboa-Schmeidler Multi-Prior Model H.2 Applications to Welfare Ec onomics 1 The Second Fundamen tal Theorem of Welfare Economics 2 Characterization of Pareto Optima 3 ∗ Harsanyi’s Utilitarianism Theorem H.3 An Application to Information Theory H.4 ∗ Applications to Financial Economics 1 Viability an d Arbitrage-Free P rice Functionals 2 The No-Arbitrage Theorem H.5 Applications to Cooperative Games 1 The Nash Bargaining Solution 2 ∗ Coalitional G ames Without S ide Payments Chapter I Metric Linear Spaces I.1 Metric Linear Spaces I.2 Cont inuous Linear Operators and Functionals vii 1 Examples of (Dis-)Continuous Linear Operators 2 Contin uity of Positive Linear Functionals 3 Closed vs . Dense Hyperplanes 4 Digression: On the Contin uit y of Concav e Functions I.3 Finite Dimensional Metric Linear Spaces I.4 ∗ Comp act Sets in Metric Linear Spaces I.5 Convex Analy sis in Metric Linea r Spaces 1 ClosureandInteriorofaConvexSet 2 Interior vs. Algebraic In terior of a Convex Set 3 Extension of Positiv e Linear Functionals, Revisited 4 Separation by Closed Hyperplanes 5 In terior vs. Algebraic Interior of a Closed and Convex Set Chapter J Normed Linear Spaces J.1 Normed Linear Spa ces 1 A G eometric Motivation 2 Normed Linear Spaces 3 Examples of Normed Linear Spaces 4 Metric vs. Normed Linear Spaces 5 Digression: The Lipschitz Continuity of Concave Maps J.2 Banach Spaces 1 Definition and Examples 2 Infinite Series in Banac h Spaces 3 ∗ On the “Size” of Banach Spaces J.3 Fixed Point T heory IV 1 The G licksberg-Fan Fixed Point T heorem 2 Application: Existence of Nash Equilibrium, Revisited 3 ∗ The Schauder Fixed Point Theorems 4 ∗ Some Consequences of Schauder’s Theorems 5 ∗ Applications to Functional Equations J.4 Bounded Linear Operators and Functionals 1 Definitions and Examples 2 Linear Homeomorphisms, Revisited 3 The Operator Norm 4 Dual Spaces 5 ∗ Discontinuous Linear Functionals, Revisited J.5 Con vex Analysis in No r med Linear Spaces 1 Separation by Closed Hyperplanes, Revisited 2 ∗ Best App roximation from a Convex Set viii 3 Extreme points J.6 Extension in Normed Linear Spaces 1 Extension of Continu ous Linear Functionals 2 ∗ Infinite Dimensional Normed Linear Spaces J.7 ∗ The Uniform Boundedness Principle Chapter K Differen tial Calculus K.1 Fréchet Differen tiation 1 Limits of Functions and Tangency 2 What is a Derivative? 3 The Fréchet Derivative 4 Examples 5 Rules of Differen tiation 6 The Second Fréchet Derivativ e of a Real Function K.2 Generalizations of the M ean Value Th eorem 1 The Generalized M ean Value Theorem 2 ∗ The Mean Value Inequality K.3 Fréchet Differentiat ion and Concave M aps 1 Remarks -on D ifferen tiability of Concave M aps 2 Fréchet Differentiable Concave Maps K.4 Optimiza tion 1 Local Extrema of R eal Maps 2 Optimization of Conca ve Maps K.5 Calculus of Variations 1 Finite Horizon Variational Problems 2 The Euler-Lagrange E quation 3 More on the Sufficiency of the Euler-Lagrange Equation 4 Infinite Horizon Variational Problems 5 Application: The Optimal Investment Problem 6 Application: The Optimal Growth Problem 7 Application: The Poincaré-Wirtinger Inequality Hin ts For Selected Exercises References Index of Symbols Index of Topics ix Preface This is primarily a textbook on mathem a tical analysis for graduate students in eco- nomics. While ther e are a large number of excellent textbooks on t his b road topic in t he mathema tics literature, m ost of these texts are overly a dvanced relative t o the needs of a vast majority of economics studen ts, and c o ncentrate on various t opics that a re not readily helpful for studying economic theory. Moreover, it seems that most economics students lack the time and/or co urage to enroll in a mat h course at the graduate lev el. Sometim es this is not even for bad rea sons, for only few math depa rtm ents offer c lasses that are designed fo r the particular needs of economists. Un- fortunately, m ore o ften th an no t, the consequent lack of mathematical background creates problems for the students at a later stage of their education since an ex- ceedingly l arge fraction of economic theory is impenetrable without some rigorous background in real analysis. The present text aims at providing a rem e dy for this inconvenient situation. My treatment is r igorous, yet selective. I prove a good number of results here, so the reader w ill ha ve plen ty of opportunity to sharpen his/her understanding of the “theorem-proof” duality, and t o work through a variety of “deep” theorems of mathem atical analysis. H owever, I take many sh ortcut s. Fo r instance, I av oid com- plex numbers at a ll c ost, assume compactness of things w hen one could get aw ay w ith separab ility, introduce topological and/or topological linear con ce p ts only via metrics and/or norms, and so on . M y objective i s not to report even the m ain theorems i n their most general form, but rather to giv e a g ood idea to the studen t w h y t hese are true, or even more im portan tly, w hy one should s uspect tha t they must be true even before t hey are proved. But the shortcuts a re not ov erly extensive in t he sense t hat the main results covered here posses s a g ood degree of app licability, especially for ma in stream economics. Indeed, the purely math ematical developm ent of the text is put to good use th rou gh sever al a pp lic ation s that provide concise introduction s to a variety o f topics from economic theory. Among these topics are individual decision theory, cooperativ e and n oncooperativ e game theory, welfare econ om ics, info rma tion theory, g eneral equilibrium and finance, and in tertemporal e conomics. An obv ious dimension that differentiates this text from various books on real analysis pertains to the choice of topics. I put much more emphasis on topics that are immediately relevant for economic theory, and omit some standard t hem es of real analysis that are o f second ary importance for economists. In partic ular , unlike most treatments of mathematical analysis found in the literature, I work here quite a bit on order theory, conv ex analy sis, optimization, linear and n o nlin e a r corres pondenc es , dynamic programming, and calculus of variations. M oreo ver, apart from direct appli- cations to economic theory, the exposition includes quite a few fixed point theorems, along with a leisurely introduction to differential calculus in Banach spaces. (Indeed, the latter half of the text can be thought of as providing a modest in troduction to x [...]... consists of four parts: I Set Theory (Chapters A-B) II Analysis on Metric Spaces (Chapters C-E) III Analysis on Linear Spaces (Chapters F-H) IV Analysis on Metric/Normed Linear Spaces (Chapters I-K) Part I provides an elementary, yet fairly comprehensive, overview of (intuitive) set theory Covering the fundamental notions of sets, relations, functions, real sequences, basic calculus, and countability,... for economic theorists You decide: “ A good economic theorist can look at a problem in more than one way In particular, a good economic theorist will ‘think like a pure theorist when doing pure economic theory and like an applied theorist when doing applied theory’ (Great economic theorists think like themselves when doing economics.)” xii forms of the Hahn-Banach Theorem is given here, along with. ..geometric (non)linear analysis. ) However, because they play only a minor role in modern economic theory, I do not at all discuss topics like Fourier analysis, Hilbert spaces and spectral theory in this book While I assume here that the student is familiar with the notion of “proof” — within the first semester of a graduate economics program, this goal must be achieved — I... Hahn-Banach Theorem is given here, along with several economic applications that range from individual decision theory to financial economics Among the most notable theorems covered are the Hahn-Banach Extension Theorem, the Krein-Rutman Theorem, and the Dieudonné Separation Theorem Part IV can be considered as a primer on geometric linear and nonlinear analysis Since I wish to avoid the consideration of... companion volume called Probability Theory with Economic Applications On Alternative Uses of the Text This book is intended to serve as a textbook for a number of different courses, and also for independent study • A second graduate course on mathematics-for-economists Such a course would use Chapter A for review, and cover the first section of Chapter B, along with pretty much all of Chapters C, D and... something like one half to two-thirds of a semester, depending on how long one wishes to spend on the applications of dynamic programming and game theory The remaining part of the semester may then be used to go deeper into a variety of fields, such as convex analysis (Chapters F—H and parts of Chapters I and J), introductory linear analysis (Chapters F-J), or introductory nonlinear analysis and fixed point... useful for complementary reading on a good number of topics that are traditionally covered in a first math-for-econ course, especially if the instructor wishes to touch upon infinite dimensional matters as well (Examples The earlier parts of Chapters C-E complements the standard coverage of real analysis within Rn , Chapter C spends quite a bit of time on the Contraction Mapping Theorem and its 4 To the... would be able to read this book with a considerably accelerated pace Similarly, after the present course, the advanced texts like Mas-Colell (1989), Duffie (1996), and Becker and Boyd III (1997) should be within reach Within the mathematics folklore, this book would be viewed as a continuation of a first “mathematical analysis course, which is usually taught after or along with xv “advanced calculus.” In... of real- to -real functions often helps getting intuition about things in more abstract settings Finally, while most students come across metric spaces by the end of the first semester of their graduate education in economics, I do not assume any prior knowledge of this topic here To judge things for yourself, check if you have some “feeling” for the following facts: • Every monotonic sequence of real. .. results, or substantial generalizations of them, are proved within the text.) The economic applications covered here are foundational for the large part, so they do not require any sophisticated economic training However, if you have taken at least one graduate course on microeconomic theory, then you would probably appreciate the importance of these applications better xvii BASIC CONVENTIONS • The frequently . Set Theory (Chapters A-B) II. Analysis on Metric Spaces (Chapters C-E) III. Analysis on Linear Spa ces (Chapters F-H) IV. Analysis on Metric/Normed Linear Spaces (Chapters I-K) Part I pro v id. . REAL ANALYSIS with ECONOMIC APPLICAT IONS EFE A. OK New York Universit y December, 2005 . mat he matics is very much like poetry what m akes a good poem - a great poem - i s that. applied theory’. (Great economic theorists think like themselves when doing economics.)” xii forms of the Hahn-Banac h Theorem is given here, along with several economic a p- plications th at range